On additive MDS codes with linear projections

We support some evidence that a long additive MDS code over a finite field must be equivalent to a linear code. More precisely, let C be an F-q-linear (n, q(hk), n - k + 1)(qh) MDS code over F-qh. If k = 3, h is an element of {2, 3}, n > max {q(h-1), hq - 1} + 3, and C has three coordinates from which its projections are equivalent to Fqh-linear codes, we prove that C itself is equivalent to an F-qh-linear code. If k > 3, n > q + k, and there are two disjoint subsets of coordinates whose combined size is at most k - 2 from which the projections of C are equivalent to F-qh-linear codes, we prove that C is equivalent to a code which is linear over a larger field than F-q. (C) 2023 Elsevier Inc. All rights reserved.